In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a field is a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
on which
addition,
subtraction,
multiplication, and
division are defined and behave as the corresponding operations on
rational and
real numbers do. A field is thus a fundamental
algebraic structure which is widely used in
algebra,
number theory, and many other areas of mathematics.
The best known fields are the field of
rational numbers, the field of
real numbers and the field of
complex numbers. Many other fields, such as
fields of rational functions,
algebraic function fields,
algebraic number fields, and
''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and
algebraic geometry. Most
cryptographic protocol
A security protocol (cryptographic protocol or encryption protocol) is an abstract or concrete protocol that performs a security-related function and applies cryptographic methods, often as sequences of cryptographic primitives. A protocol descri ...
s rely on
finite fields, i.e., fields with finitely many
elements.
The relation of two fields is expressed by the notion of a
field extension.
Galois theory, initiated by
Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, this theory shows that
angle trisection and
squaring the circle cannot be done with a
compass and straightedge. Moreover, it shows that
quintic equations are, in general, algebraically unsolvable.
Fields serve as foundational notions in several mathematical domains. This includes different branches of
mathematical analysis, which are based on fields with additional structure. Basic theorems in analysis hinge on the structural properties of the field of real numbers. Most importantly for algebraic purposes, any field may be used as the
scalars
Scalar may refer to:
* Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
* Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
for a
vector space, which is the standard general context for
linear algebra.
Number fields, the siblings of the field of rational numbers, are studied in depth in
number theory.
Function fields can help describe properties of geometric objects.
Definition
Informally, a field is a set, along with two
operation
Operation or Operations may refer to:
Arts, entertainment and media
* ''Operation'' (game), a battery-operated board game that challenges dexterity
* Operation (music), a term used in musical set theory
* ''Operations'' (magazine), Multi-Man ...
s defined on that set: an addition operation written as , and a multiplication operation written as , both of which behave similarly as they behave for
rational numbers and
real numbers, including the existence of an
additive inverse for all elements , and of a
multiplicative inverse for every nonzero element . This allows one to also consider the so-called ''inverse'' operations of subtraction, , and division, , by defining:
:,
:.
Classic definition
Formally, a field is a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
together with two
binary operations on called ''addition'' and ''multiplication''. A binary operation on is a mapping , that is, a correspondence that associates with each ordered pair of elements of a uniquely determined element of . The result of the addition of and is called the sum of and , and is denoted . Similarly, the result of the multiplication of and is called the product of and , and is denoted or . These operations are required to satisfy the following properties, referred to as ''
field axioms'' (in these axioms, , , and are arbitrary
elements of the field ):
*
Associativity of addition and multiplication: , and .
*
Commutativity of addition and multiplication: , and .
*
Additive and
multiplicative identity: there exist two different elements and in such that and .
*
Additive inverses: for every in , there exists an element in , denoted , called the ''additive inverse'' of , such that .
*
Multiplicative inverses: for every in , there exists an element in , denoted by or , called the ''multiplicative inverse'' of , such that .
*
Distributivity of multiplication over addition: .
This may be summarized by saying: a field has two operations, called addition and multiplication; it is an
abelian group under addition with 0 as the additive identity; the nonzero elements are an abelian group under multiplication with 1 as the multiplicative identity; and multiplication distributes over addition.
Even more summarized: a field is a
commutative ring where
and all nonzero elements are
invertible under multiplication.
Alternative definition
Fields can also be defined in different, but equivalent ways. One can alternatively define a field by four binary operations (addition, subtraction, multiplication, and division) and their required properties.
Division by zero is, by definition, excluded. In order to avoid
existential quantifiers, fields can be defined by two binary operations (addition and multiplication), two unary operations (yielding the additive and multiplicative inverses respectively), and two
nullary operations (the constants and ). These operations are then subject to the conditions above. Avoiding existential quantifiers is important in
constructive mathematics and
computing
Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes, and development of both hardware and software. Computing has scientific, ...
. One may equivalently define a field by the same two binary operations, one unary operation (the multiplicative inverse), and two constants and , since and .
[The a priori twofold use of the symbol "−" for denoting one part of a constant and for the additive inverses is justified by this latter condition.]
Examples
Rational numbers
Rational numbers have been widely used a long time before the elaboration of the concept of field.
They are numbers that can be written as
fractions
, where and are
integers, and . The additive inverse of such a fraction is , and the multiplicative inverse (provided that ) is , which can be seen as follows:
:
The abstractly required field axioms reduce to standard properties of rational numbers. For example, the law of distributivity can be proven as follows:
:
Real and complex numbers
The
real numbers , with the usual operations of addition and multiplication, also form a field. The
complex numbers consist of expressions
: with real,
where is the
imaginary unit, i.e., a (non-real) number satisfying .
Addition and multiplication of real numbers are defined in such a way that expressions of this type satisfy all field axioms and thus hold for . For example, the distributive law enforces
:
It is immediate that this is again an expression of the above type, and so the complex numbers form a field. Complex numbers can be geometrically represented as points in the
plane, with
Cartesian coordinates given by the real numbers of their describing expression, or as the arrows from the origin to these points, specified by their length and an angle enclosed with some distinct direction. Addition then corresponds to combining the arrows to the intuitive parallelogram (adding the Cartesian coordinates), and the multiplication is – less intuitively – combining rotating and scaling of the arrows (adding the angles and multiplying the lengths). The fields of real and complex numbers are used throughout mathematics, physics, engineering, statistics, and many other scientific disciplines.
Constructible numbers
In antiquity, several geometric problems concerned the (in)feasibility of constructing certain numbers with
compass and straightedge. For example, it was unknown to the Greeks that it is, in general, impossible to trisect a given angle in this way. These problems can be settled using the field of
constructible numbers. Real constructible numbers are, by definition, lengths of line segments that can be constructed from the points 0 and 1 in finitely many steps using only
compass and
straightedge. These numbers, endowed with the field operations of real numbers, restricted to the constructible numbers, form a field, which properly includes the field of rational numbers. The illustration shows the construction of
square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
...
s of constructible numbers, not necessarily contained within . Using the labeling in the illustration, construct the segments , , and a
semicircle over (center at the
midpoint ), which intersects the
perpendicular line through in a point , at a distance of exactly
from when has length one.
Not all real numbers are constructible. It can be shown that
is not a constructible number, which implies that it is impossible to construct with compass and straightedge the length of the side of a
cube with volume 2, another problem posed by the ancient Greeks.
A field with four elements
In addition to familiar number systems such as the rationals, there are other, less immediate examples of fields. The following example is a field consisting of four elements called , , , and . The notation is chosen such that plays the role of the additive identity element (denoted 0 in the axioms above), and is the multiplicative identity (denoted 1 in the axioms above). The field axioms can be verified by using some more field theory, or by direct computation. For example,
: , which equals , as required by the distributivity.
This field is called a
finite field with four elements, and is denoted or . The
subset consisting of and (highlighted in red in the tables at the right) is also a field, known as the ''
binary field'' or . In the context of
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
and
Boolean algebra, and are often denoted respectively by ''false'' and ''true'', and the addition is then denoted
XOR (exclusive or). In other words, the structure of the binary field is the basic structure that allows computing with
bits.
Elementary notions
In this section, denotes an arbitrary field and and are arbitrary
elements of .
Consequences of the definition
One has and . In particular, one may deduce the additive inverse of every element as soon as one knows .
If then or must be 0, since, if , then
. This means that every field is an
integral domain.
In addition, the following properties are true for any elements and :
:
:
:
:
: if
The additive and the multiplicative group of a field
The axioms of a field imply that it is an
abelian group under addition. This group is called the
additive group of the field, and is sometimes denoted by when denoting it simply as could be confusing.
Similarly, the ''nonzero'' elements of form an abelian group under multiplication, called the
multiplicative group, and denoted by or just or .
A field may thus be defined as set equipped with two operations denoted as an addition and a multiplication such that is an abelian group under addition, is an abelian group under multiplication (where 0 is the identity element of the addition), and multiplication is
distributive over addition.
[Equivalently, a field is an algebraic structure of type , such that is not defined, and
are abelian groups, and ⋅ is distributive over +. ] Some elementary statements about fields can therefore be obtained by applying general facts of
groups
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
. For example, the additive and multiplicative inverses and are uniquely determined by .
The requirement follows, because 1 is the identity element of a group that does not contain 0. Thus, the
trivial ring, consisting of a single element, is not a field.
Every finite
subgroup of the multiplicative group of a field is
cyclic
Cycle, cycles, or cyclic may refer to:
Anthropology and social sciences
* Cyclic history, a theory of history
* Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr.
* Social cycle, various cycles in so ...
(see ).
Characteristic
In addition to the multiplication of two elements of ''F'', it is possible to define the product of an arbitrary element of by a positive
integer to be the -fold sum
: (which is an element of .)
If there is no positive integer such that
:,
then is said to have
characteristic 0. For example, the field of rational numbers has characteristic 0 since no positive integer is zero. Otherwise, if there ''is'' a positive integer satisfying this equation, the smallest such positive integer can be shown to be a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
. It is usually denoted by and the field is said to have characteristic then.
For example, the field has characteristic 2 since (in the notation of the above addition table) .
If has characteristic , then for all in . This implies that
:,
since all other
binomial coefficients appearing in the
binomial formula are divisible by . Here, ( factors) is the -th power, i.e., the -fold product of the element . Therefore, the
Frobenius map
:
is compatible with the addition in (and also with the multiplication), and is therefore a field homomorphism. The existence of this homomorphism makes fields in characteristic quite different from fields of characteristic 0.
Subfields and prime fields
A ''
subfield'' of a field is a subset of that is a field with respect to the field operations of . Equivalently is a subset of that contains , and is closed under addition, multiplication, additive inverse and multiplicative inverse of a nonzero element. This means that , that for all both and are in , and that for all in , both and are in .
Field homomorphism
Field theory is the branch of mathematics in which fields are studied. This is a glossary of some terms of the subject. (See field theory (physics) for the unrelated field theories in physics.)
Definition of a field
A field is a commutative rin ...
s are maps between two fields such that , , and , where and are arbitrary elements of . All field homomorphisms are
injective. If is also
surjective, it is called an
isomorphism (or the fields and are called isomorphic).
A field is called a
prime field if it has no proper (i.e., strictly smaller) subfields. Any field contains a prime field. If the characteristic of is (a prime number), the prime field is isomorphic to the finite field introduced below. Otherwise the prime field is isomorphic to .
Finite fields
''Finite fields'' (also called ''Galois fields'') are fields with finitely many elements, whose number is also referred to as the order of the field. The above introductory example is a field with four elements. Its subfield is the smallest field, because by definition a field has at least two distinct elements .
The simplest finite fields, with prime order, are most directly accessible using
modular arithmetic. For a fixed positive integer , arithmetic "modulo " means to work with the numbers
:
The addition and multiplication on this set are done by performing the operation in question in the set of integers, dividing by and taking the remainder as result. This construction yields a field precisely if is a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
. For example, taking the prime results in the above-mentioned field . For and more generally, for any
composite number
A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor
In mathematics, a divisor of an integer n, also called a factor ...
(i.e., any number which can be expressed as a product of two strictly smaller natural numbers), is not a field: the product of two non-zero elements is zero since in , which, as was explained
above, prevents from being a field. The field with elements ( being prime) constructed in this way is usually denoted by .
Every finite field has elements, where is prime and . This statement holds since may be viewed as a
vector space over its prime field. The
dimension of this vector space is necessarily finite, say , which implies the asserted statement.
A field with elements can be constructed as the
splitting field of the
polynomial
:.
Such a splitting field is an extension of in which the polynomial has zeros. This means has as many zeros as possible since the
degree of is . For , it can be checked case by case using the above multiplication table that all four elements of satisfy the equation , so they are zeros of . By contrast, in , has only two zeros (namely 0 and 1), so does not split into linear factors in this smaller field. Elaborating further on basic field-theoretic notions, it can be shown that two finite fields with the same order are isomorphic. It is thus customary to speak of ''the'' finite field with elements, denoted by or .
History
Historically, three algebraic disciplines led to the concept of a field: the question of solving polynomial equations,
algebraic number theory, and
algebraic geometry. A first step towards the notion of a field was made in 1770 by
Joseph-Louis Lagrange, who observed that permuting the zeros of a
cubic polynomial in the expression
:
(with being a third
root of unity) only yields two values. This way, Lagrange conceptually explained the classical solution method of
Scipione del Ferro and
François Viète, which proceeds by reducing a cubic equation for an unknown to a quadratic equation for . Together with a similar observation for
equations of degree 4, Lagrange thus linked what eventually became the concept of fields and the concept of groups.
Vandermonde, also in 1770, and to a fuller extent,
Carl Friedrich Gauss, in his ''
Disquisitiones Arithmeticae'' (1801), studied the equation
:
for a prime and, again using modern language, the resulting cyclic
Galois group. Gauss deduced that a
regular -gon can be constructed if . Building on Lagrange's work,
Paolo Ruffini claimed (1799) that
quintic equations (polynomial equations of degree 5) cannot be solved algebraically; however, his arguments were flawed. These gaps were filled by
Niels Henrik Abel in 1824.
Évariste Galois, in 1832, devised necessary and sufficient criteria for a polynomial equation to be algebraically solvable, thus establishing in effect what is known as
Galois theory today. Both Abel and Galois worked with what is today called an
algebraic number field, but conceived neither an explicit notion of a field, nor of a group.
In 1871
Richard Dedekind introduced, for a set of real or complex numbers that is closed under the four arithmetic operations, the
German
German(s) may refer to:
* Germany (of or related to)
**Germania (historical use)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
** For citizens of Germany, see also German nationality law
**Ge ...
word ''Körper'', which means "body" or "corpus" (to suggest an organically closed entity). The English term "field" was introduced by .
In 1881
Leopold Kronecker defined what he called a ''domain of rationality'', which is a field of
rational fractions in modern terms. Kronecker's notion did not cover the field of all algebraic numbers (which is a field in Dedekind's sense), but on the other hand was more abstract than Dedekind's in that it made no specific assumption on the nature of the elements of a field. Kronecker interpreted a field such as abstractly as the rational function field . Prior to this, examples of transcendental numbers were known since
Joseph Liouville's work in 1844, until
Charles Hermite
Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra.
...
(1873) and
Ferdinand von Lindemann (1882) proved the transcendence of and , respectively.
The first clear definition of an abstract field is due to . In particular,
Heinrich Martin Weber's notion included the field F
''p''.
Giuseppe Veronese (1891) studied the field of formal power series, which led to introduce the field of ''p''-adic numbers. synthesized the knowledge of abstract field theory accumulated so far. He axiomatically studied the properties of fields and defined many important field-theoretic concepts. The majority of the theorems mentioned in the sections
Galois theory,
Constructing fields and
Elementary notions can be found in Steinitz's work. linked the notion of
orderings in a field, and thus the area of analysis, to purely algebraic properties.
Emil Artin redeveloped Galois theory from 1928 through 1942, eliminating the dependency on the
primitive element theorem.
Constructing fields
Constructing fields from rings
A
commutative ring is a set, equipped with an addition and multiplication operation, satisfying all the axioms of a field, except for the existence of multiplicative inverses . For example, the integers form a commutative ring, but not a field: the
reciprocal
Reciprocal may refer to:
In mathematics
* Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal''
* Reciprocal polynomial, a polynomial obtained from another pol ...
of an integer is not itself an integer, unless .
In the hierarchy of algebraic structures fields can be characterized as the commutative rings in which every nonzero element is a
unit (which means every element is invertible). Similarly, fields are the commutative rings with precisely two distinct
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...
s, and . Fields are also precisely the commutative rings in which is the only
prime ideal.
Given a commutative ring , there are two ways to construct a field related to , i.e., two ways of modifying such that all nonzero elements become invertible: forming the field of fractions, and forming residue fields. The field of fractions of is , the rationals, while the residue fields of are the finite fields .
Field of fractions
Given an
integral domain , its
field of fractions is built with the fractions of two elements of exactly as Q is constructed from the integers. More precisely, the elements of are the fractions where and are in , and . Two fractions and are equal if and only if . The operation on the fractions work exactly as for rational numbers. For example,
:
It is straightforward to show that, if the ring is an integral domain, the set of the fractions form a field.
The field of the
rational fractions over a field (or an integral domain) is the field of fractions of the
polynomial ring . The field of
Laurent series
:
over a field is the field of fractions of the ring of
formal power series (in which ). Since any Laurent series is a fraction of a power series divided by a power of (as opposed to an arbitrary power series), the representation of fractions is less important in this situation, though.
Residue fields
In addition to the field of fractions, which embeds
injectively into a field, a field can be obtained from a commutative ring by means of a
surjective map onto a field . Any field obtained in this way is a
quotient , where is a
maximal ideal of . If
has only one maximal ideal , this field is called the
residue field of .
The
ideal generated by a single polynomial in the polynomial ring (over a field ) is maximal if and only if is
irreducible in , i.e., if cannot be expressed as the product of two polynomials in of smaller
degree. This yields a field
:
This field contains an element (namely the
residue class of ) which satisfies the equation
:.
For example, is obtained from by
adjoining the
imaginary unit symbol , which satisfies , where . Moreover, is irreducible over , which implies that the map that sends a polynomial to yields an isomorphism
:
Constructing fields within a bigger field
Fields can be constructed inside a given bigger container field. Suppose given a field , and a field containing as a subfield. For any element of , there is a smallest subfield of containing and , called the subfield of ''F'' generated by and denoted . The passage from to is referred to by ''
adjoining an element'' to . More generally, for a subset , there is a minimal subfield of containing and , denoted by .
The
compositum
In mathematics, the tensor product of two fields is their tensor product as algebras over a common subfield. If no subfield is explicitly specified, the two fields must have the same characteristic and the common subfield is their prime subf ...
of two subfields and of some field is the smallest subfield of containing both and The compositum can be used to construct the biggest subfield of satisfying a certain property, for example the biggest subfield of , which is, in the language introduced below, algebraic over .
[Further examples include the maximal unramified extension or the maximal abelian extension within .]
Field extensions
The notion of a subfield can also be regarded from the opposite point of view, by referring to being a ''
field extension'' (or just extension) of , denoted by
:,
and read " over ".
A basic datum of a field extension is its
degree , i.e., the dimension of as an -vector space. It satisfies the formula
:.
Extensions whose degree is finite are referred to as finite extensions. The extensions and are of degree 2, whereas is an infinite extension.
Algebraic extensions
A pivotal notion in the study of field extensions are
algebraic element
In mathematics, if is a field extension of , then an element of is called an algebraic element over , or just algebraic over , if there exists some non-zero polynomial with coefficients in such that . Elements of which are not algebraic ove ...
s. An element is ''algebraic'' over if it is a
root of a
polynomial with
coefficients in , that is, if it satisfies a
polynomial equation
:,
with in , and .
For example, the
imaginary unit in is algebraic over , and even over , since it satisfies the equation
:.
A field extension in which every element of is algebraic over is called an
algebraic extension. Any finite extension is necessarily algebraic, as can be deduced from the above multiplicativity formula.
The subfield generated by an element , as above, is an algebraic extension of if and only if is an algebraic element. That is to say, if is algebraic, all other elements of are necessarily algebraic as well. Moreover, the degree of the extension , i.e., the dimension of as an -vector space, equals the minimal degree such that there is a polynomial equation involving , as above. If this degree is , then the elements of have the form
:
For example, the field of
Gaussian rationals is the subfield of consisting of all numbers of the form where both and are rational numbers: summands of the form (and similarly for higher exponents) don't have to be considered here, since can be simplified to .
Transcendence bases
The above-mentioned field of
rational fractions , where is an
indeterminate, is not an algebraic extension of since there is no polynomial equation with coefficients in whose zero is . Elements, such as , which are not algebraic are called
transcendental. Informally speaking, the indeterminate and its powers do not interact with elements of . A similar construction can be carried out with a set of indeterminates, instead of just one.
Once again, the field extension discussed above is a key example: if is not algebraic (i.e., is not a
root of a polynomial with coefficients in ), then is isomorphic to . This isomorphism is obtained by substituting to in rational fractions.
A subset of a field is a
transcendence basis
In abstract algebra, the transcendence degree of a field extension ''L'' / ''K'' is a certain rather coarse measure of the "size" of the extension. Specifically, it is defined as the largest cardinality of an algebraically independent subset o ...
if it is
algebraically independent (don't satisfy any polynomial relations) over and if is an algebraic extension of . Any field extension has a transcendence basis. Thus, field extensions can be split into ones of the form (
purely transcendental extensions) and algebraic extensions.
Closure operations
A field is
algebraically closed if it does not have any strictly bigger algebraic extensions or, equivalently, if any
polynomial equation
:, with coefficients ,
has a solution . By the
fundamental theorem of algebra, is algebraically closed, i.e., ''any'' polynomial equation with complex coefficients has a complex solution. The rational and the real numbers are ''not'' algebraically closed since the equation
:
does not have any rational or real solution. A field containing is called an ''
algebraic closure'' of if it is
algebraic
Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings.
Algebraic may also refer to:
* Algebraic data type, a data ...
over (roughly speaking, not too big compared to ) and is algebraically closed (big enough to contain solutions of all polynomial equations).
By the above, is an algebraic closure of . The situation that the algebraic closure is a finite extension of the field is quite special: by the
Artin-Schreier theorem, the degree of this extension is necessarily 2, and is
elementarily equivalent to . Such fields are also known as
real closed fields.
Any field has an algebraic closure, which is moreover unique up to (non-unique) isomorphism. It is commonly referred to as ''the'' algebraic closure and denoted . For example, the algebraic closure of is called the field of
algebraic numbers. The field is usually rather implicit since its construction requires the
ultrafilter lemma, a set-theoretic axiom that is weaker than the
axiom of choice. In this regard, the algebraic closure of , is exceptionally simple. It is the union of the finite fields containing (the ones of order ). For any algebraically closed field of characteristic 0, the algebraic closure of the field of
Laurent series is the field of
Puiseux series, obtained by adjoining roots of .
Fields with additional structure
Since fields are ubiquitous in mathematics and beyond, several refinements of the concept have been adapted to the needs of particular mathematical areas.
Ordered fields
A field ''F'' is called an ''ordered field'' if any two elements can be compared, so that and whenever and . For example, the real numbers form an ordered field, with the usual ordering . The
Artin-Schreier theorem states that a field can be ordered if and only if it is a
formally real field, which means that any quadratic equation
:
only has the solution . The set of all possible orders on a fixed field is isomorphic to the set of
ring homomorphisms from the
Witt ring of
quadratic forms over , to .
An
Archimedean field is an ordered field such that for each element there exists a finite expression
:
whose value is greater than that element, that is, there are no infinite elements. Equivalently, the field contains no
infinitesimals
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally refe ...
(elements smaller than all rational numbers); or, yet equivalent, the field is isomorphic to a subfield of .
An ordered field is
Dedekind-complete if all
upper bounds,
lower bound
In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of .
Dually, a lower bound or minorant of is defined to be an elemen ...
s (see
Dedekind cut) and limits, which should exist, do exist. More formally, each
bounded subset
:''"Bounded" and "boundary" are distinct concepts; for the latter see boundary (topology). A circle in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary.
In mathematical analysis and related areas of mat ...
of is required to have a least upper bound. Any complete field is necessarily Archimedean, since in any non-Archimedean field there is neither a greatest infinitesimal nor a least positive rational, whence the sequence , every element of which is greater than every infinitesimal, has no limit.
Since every proper subfield of the reals also contains such gaps, is the unique complete ordered field, up to isomorphism. Several foundational results in
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
follow directly from this characterization of the reals.
The
hyperreals
In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains number ...
form an ordered field that is not Archimedean. It is an extension of the reals obtained by including infinite and infinitesimal numbers. These are larger, respectively smaller than any real number. The hyperreals form the foundational basis of
non-standard analysis.
Topological fields
Another refinement of the notion of a field is a
topological field, in which the set is a
topological space, such that all operations of the field (addition, multiplication, the maps and ) are
continuous maps with respect to the topology of the space.
The topology of all the fields discussed below is induced from a
metric, i.e., a
function
:
that measures a ''distance'' between any two elements of .
The
completion of is another field in which, informally speaking, the "gaps" in the original field are filled, if there are any. For example, any
irrational number , such as , is a "gap" in the rationals in the sense that it is a real number that can be approximated arbitrarily closely by rational numbers , in the sense that distance of and given by the
absolute value is as small as desired.
The following table lists some examples of this construction. The fourth column shows an example of a zero
sequence, i.e., a sequence whose limit (for ) is zero.
The field is used in number theory and
-adic analysis. The algebraic closure carries a unique norm extending the one on , but is not complete. The completion of this algebraic closure, however, is algebraically closed. Because of its rough analogy to the complex numbers, it is sometimes called the field of
complex p-adic numbers and is denoted by .
Local fields
The following topological fields are called ''
local fields'':
[Some authors also consider the fields and to be local fields. On the other hand, these two fields, also called Archimedean local fields, share little similarity with the local fields considered here, to a point that calls them "completely anomalous".]
* finite extensions of (local fields of characteristic zero)
* finite extensions of , the field of Laurent series over (local fields of characteristic ).
These two types of local fields share some fundamental similarities. In this relation, the elements and (referred to as
uniformizer) correspond to each other. The first manifestation of this is at an elementary level: the elements of both fields can be expressed as power series in the uniformizer, with coefficients in . (However, since the addition in is done using
carrying, which is not the case in , these fields are not isomorphic.) The following facts show that this superficial similarity goes much deeper:
* Any
first-order
In mathematics and other formal sciences, first-order or first order most often means either:
* "linear" (a polynomial of degree at most one), as in first-order approximation and other calculus uses, where it is contrasted with "polynomials of hig ...
statement that is true for almost all is also true for almost all . An application of this is the
Ax-Kochen theorem describing zeros of homogeneous polynomials in .
*
Tamely ramified extensions of both fields are in bijection to one another.
* Adjoining arbitrary -power roots of (in ), respectively of (in ), yields (infinite) extensions of these fields known as
perfectoid fields. Strikingly, the Galois groups of these two fields are isomorphic, which is the first glimpse of a remarkable parallel between these two fields:
Differential fields
Differential fields are fields equipped with a
derivation, i.e., allow to take derivatives of elements in the field. For example, the field R(''X''), together with the standard derivative of polynomials forms a differential field. These fields are central to
differential Galois theory, a variant of Galois theory dealing with
linear differential equations.
Galois theory
Galois theory studies
algebraic extensions of a field by studying the
symmetry in the arithmetic operations of addition and multiplication. An important notion in this area is that of
finite Galois extensions , which are, by definition, those that are
separable and
normal. The
primitive element theorem shows that finite separable extensions are necessarily
simple
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnn ...
, i.e., of the form
:,
where is an irreducible polynomial (as above). For such an extension, being normal and separable means that all zeros of are contained in and that has only simple zeros. The latter condition is always satisfied if has characteristic 0.
For a finite Galois extension, the
Galois group is the group of
field automorphisms of that are trivial on (i.e., the
bijections that preserve addition and multiplication and that send elements of to themselves). The importance of this group stems from the
fundamental theorem of Galois theory
In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions in relation to groups. It was proved by Évariste Galois in his development of Galois theory.
In its most basi ...
, which constructs an explicit
one-to-one correspondence between the set of
subgroups of and the set of intermediate extensions of the extension . By means of this correspondence, group-theoretic properties translate into facts about fields. For example, if the Galois group of a Galois extension as above is not
solvable (cannot be built from
abelian groups), then the zeros of ''cannot'' be expressed in terms of addition, multiplication, and radicals, i.e., expressions involving